てたまらない: an almost unbearable sensation or desire
Express intense heat, pain, anticipation, or desire. Distinguish connecting forms, the direction of the feeling, and whether a following action actually occurred.
A very intense sensation
暑くてたまりません。 describes heat that is hard to bear. It uses the i-adjective connective 暑くて and expresses much greater felt intensity than being slightly warm.
足が痛くてたまらなかったので、座って休みました。 means that the speaker's feet hurt intensely, so they sat down and rested. The sentence reports both sitting and resting, not merely an intention to rest.
Positive feelings and desires too
旅行が楽しみでたまりません。 expresses intense anticipation of a trip. 楽しみだ connects with で as a na-adjective expression; たまらない does not require an unpleasant feeling.
友達に会いたくてたまりません。 expresses a very strong wish to meet a friend. 会いたい becomes 会いたくて, so the sentence does not report that the meeting has already happened.
Intensity differs from permission or prohibition
ここで遊んではいけません。 prohibits playing here. 遊びたくてたまりません。 describes an intense desire to play, leaving permission as a separate issue.
眠くてしょうがない and 眠くてたまらない can both describe extreme sleepiness. Learn combinations suited to strong sensations, emotions, and desires rather than treating this as a formula attachable to every verb.
Quick check: Does 合格して、うれしくてたまりません。 mean that the speaker failed?
Show the answer and why
No. It means the speaker passed and is extremely happy. たまらない intensifies the happiness here; it denies neither passing nor happiness.